Home›Calculus 3›Quadric surfaces in standard form›Problem 9.731

Quadric surfaces in standard form

Problem 9.731 · easy

Write \( \displaystyle - 27 x^{2} + 75 y^{2} - 675 z = 0 \) in standard form and identify the surface.

Divide through by 675 and solve for the variable to the first power.
Reviewed
\[ \frac{- 27 x^{2} + 75 y^{2} - 675 z}{675} = - \frac{x^{2}}{25} + \frac{y^{2}}{9} - z \]
The standard form.✓ Proved
A hyperbolic paraboloid: z to the first power equals a difference of squares.
Reviewed

Answer \( - \frac{x^{2}}{25} + \frac{y^{2}}{9} - z = 0\ (\text{hyperbolic paraboloid}) \)

Lines: 1 proved, 2 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	a sentence; read, not computed
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the surface classified from the signs of its coefficients; the standard form is a multiple of the given equation

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: pass — The solution correctly derives the standard form and accurately identifies the surface based on the resulting equation structure.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly derives the standard form and accurately identifies the surface based on the resulting equation structure.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-16 — The solution correctly derives the standard form and accurately identifies the surface based on the equation structure.
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: pass 2026-10-15 — The solution correctly divides by 675 to isolate z, resulting in the standard form z = y^2/9 - x^2/25. The identification of the surface as a hyperbolic paraboloid is correct.
gpt-oss:20b: fail (misleading) 2026-10-15 — The first sentence claims to solve for the variable to the first power, but the step only divides by 675; it does not isolate z. The statement is therefore misleading.

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/quadric_surfaces, checked 2026-10-18 with SymPy 1.14.0.